zbMATH — the first resource for mathematics

Modulations of Sinh-Gordon and Sine-Gordon wavetrains. (English) Zbl 0541.35071
An invariant representation of the modulation equations for the sinh- and sine-Gordon wavetrains is derived. A simple derivation of the representation which makes fundamental use of squared eigenfunctions is presented. This representation is used to place the modulation equations in Riemann invariant form and to cast them in a Hamiltonian form. The multiphase sinh-Gordon study is complete, while the sine-Gordon theory for more than one phase possesses technical difficulties which are described in the text. Explicit results on real two-phase sine-Gordon waves are included in Section VI.

35Q99 Partial differential equations of mathematical physics and other areas of application
35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
Full Text: DOI
[1] Ablowitz, The evolution of multi-phase modes for nonlinear dispersive waves, Stud. Appl. Math. 49 pp 225– (1970) · Zbl 0203.41001 · doi:10.1002/sapm1970493225
[2] Flaschka, Multiphase averaging and the inverse spectral solution of the Korteweg-deVries equation, Comm. Pure Appl. Math. 33 pp 739– (1980) · Zbl 0454.35080 · doi:10.1002/cpa.3160330605
[3] McLaughlin, Modulations of KdV Wavetrains, Physica 3 pp 335– (1981) · Zbl 1194.35377
[4] Whitham, Non-linear dispersive waves, Proc. Roy. Soc. London Ser. A 283 pp 238– (1965) · Zbl 0125.44202 · doi:10.1098/rspa.1965.0019
[5] J. Math. Phys.
[6] Forest, Canonical variables for the periodic sine-Gordon equation and a method of averaging, LA-UR 78-3318, Report of the Los Alamos Scientific Laboratory, J. Math. Phys. (1978)
[7] Takhatajian, Essentially non-linear one dimensional model of classical field theory, Theoret. and Math. Phys. 21 pp 1046– (1974) · Zbl 0299.35063 · doi:10.1007/BF01035551
[8] DatĂ©, Periodic multi-soliton solutions of the Korteweg-deVries equation and Toda lattice, Progr. Theoret. Phys. Suppl. 59 pp 107– (1976) · doi:10.1143/PTPS.59.107
[9] Dubrovin, Nonlinear equations of Korteweg-deVries type, finite-zoned linear operators, and Abelian varieties, Uspehi Mat. Nauk 31 pp 55– (1976) · Zbl 0346.35025
[10] Siegel, Topics in Complex Function Theory II (1973)
[13] McKean, The sine-Gordon and sinh-Gordon equations on the circle, Comm. Pure Appl. Math. 34 pp 197– (1981) · Zbl 0467.35078 · doi:10.1002/cpa.3160340204
[14] Whitham, Linear and Nonlinear Waves (1974)
[15] Flanders, Differential Forms with Applications to the Physical Sciences (1963) · Zbl 0112.32003
[16] Hayes, Group velocity and nonlinear dispersive wave propagation, Proc. Roy. Soc. London Ser. A 332 pp 199– (1973) · Zbl 0271.76006 · doi:10.1098/rspa.1973.0021
[17] Forest, Doctoral thesis (1979)
[18] Lamb, Elements of Soliton Theory (1979)
[19] Ablowitz, J. Math. Phys. 15 pp 11852– (1974) · doi:10.1063/1.1666551
[20] Ablowitz, Phys. Rev. Lett. (1973)
[21] Miura, SIAM J. Appl. Math. 26 pp 376– (1974) · Zbl 0273.35055 · doi:10.1137/0126036
[22] Luke, Proc. Roy. Soc. London Ser. A 292 (1966) · Zbl 0143.13603 · doi:10.1098/rspa.1966.0142
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.